Mathematics - 2014-09-04 (page 3 of 4)

V-Moy

7:00 PM

Okay, I leave chatroom now. Nightly nite all ♥(ˆ⌣ˆԅ)

Mike Miller

@Nick What do you mean by matrix variables?

Huy

Good night!

Hippalectryon

@V-Moy Nigh

Nick

@V-Moy: Nighty night.

Khallil

Do you have a picture or something that we can refer to, @Nick?

Peace out, @V-Moy.

Nick

7:00 PM

@MikeMiller: $$A = \left[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \right] $$

Most people like a different font for A

Hippalectryon

@JasperLoy Maybe you'll prefer that one :P

4

Khallil

\mathbf{A}, @Nick.

Or at least that's what I think you're referring to.

Jasper Loy

@Hippalectryon I hope nobody flags you, lol.

Daniel Fischer

@Nick Are you sure about "most"?

Hippalectryon

:c

Nick

7:02 PM

@DanielFischer: No, just a few really influential math guys.

Hippalectryon

@JasperLoy Does anybody flag in the chat ?

Khallil

Hahahaha! You da man, @Hippa!

Nick

@Hippalectryon: I did once. I will never again do it.

Jasper Loy

@Hippalectryon If you mean people flagging chat messages, yes.

Balarka Sen

Oops wrong room. Anyone knows where the mathematics chat is?

Daniel Fischer

7:03 PM

@Nick Such as? I had the impression it's mostly school teachers doing that.

Jasper Loy

Most math teachers here are quite lousy IMAO.

Nick

@DanielFischer: ... I can't be forced to remember the usernames of the users I meet in this room.

Hippalectryon

@BalarkaSen Look in English language users ;)

Khallil

Was it $\mathbf{A}$, @Nick?

Nick

@DanielFischer: I found the font \mathbb . That's what that user used when he explained some really gnarly concepts about how matrices can be used to transform stuff.

Khallil

7:07 PM

I think I've been put on ignore by @Nick. >_>

Nick

@Khallil: I don't think it matters since no one thinks there's any specific font.

Daniel Fischer

@Nick Most people reserve that font for sets.

Balarka Sen

@Chris'ssis

Khallil

Oh, I see. Yea, as @Daniel said, it's reserved for sets like $\mathbb{R}$ (real numbers), $\mathbb{C}$ (complex numbers), $\mathbb{Q}$ (rational numbers) etc.

Nick

@Khallil: I chat from a phone. I'm trying to make zero spelling mistakes and am constantly battling autocorrect. It's hard to talk. Belive me, I'm not ignoring you.

Jasper Loy

7:10 PM

Internet on phones should be discontinued, lol

Daniel Fischer

@Nick Isn't there a way to turn off autocorrection?

Balarka Sen

@Khallil I have a problem for you.

Nick

@DanielFischer: Yes, but then that'll also stop the much needed auto completion feature that comes along with it.

Khallil

What's the problem to do with, @BalarkaSen?

Balarka Sen

@Khallil Counting.

Nick

7:11 PM

@Khallil: What's the Latex code to make boxes?

Khallil

\boxed{}

Daniel Fischer

@Nick Ah, the choice between Scylla and Charybdis.

Balarka Sen

@Nick $\boxed{a}$

Khallil

Forget that, @BalarkaSen.

Counting isn't at the top of my list.

Balarka Sen

I thought you liked counting, @Khallil?

Khallil

7:12 PM

I need to watch the latest Naruto episode, @BalarkaSen.

Nick

@DanielFischer: All I heard was the choice between gibberish and gibberish. Care to explain?

Balarka Sen

You should really choose one between math and naruto, @Khallil

Khallil

Do you want to have a rematch in FIFA later on, @Huy?

I've made my choice, @BalarkaSen.

Huy

@Khallil: Can do maybe, if the time's right.

Balarka Sen

Forget about math.

Khallil

7:13 PM

In exactly 30 minutes and onwards, I can play, @Huy. ^_^

Huy

Oki. Will see.

Daniel Fischer

Between Scylla and Charybdis

Being between Scylla and Charybdis is an idiom deriving from Greek mythology, meaning "having to choose between two evils". Several other idioms, such as "on the horns of a dilemma", "between the devil and the deep blue sea", and "between a rock and a hard place" express the same meaning. == The myth and the proverb == Scylla and Charybdis were mythical sea monsters noted by Homer; later Greek tradition sited them on opposite sides of the Strait of Messina between Sicily and the Italian mainland. Scylla was rationalized as a rock shoal (described as a six-headed sea monster) on the Italian side...

Nick

@BalarkaSen: Thanks. I forget silly things... and important things. I forget all things. No more questions.

Khallil

I've forgotten about math, @BalarkaSen.

Chris's sis

@BalarkaSen Yes?

Balarka Sen

7:14 PM

@Chris'ssis Someone incredibly proved the double sum limit problem. Want to see?

Chris's sis

@BalarkaSen Which limit? Yeah, sure.

Balarka Sen

It was purely incredible.

@Chris'ssis $$\lim_{n \to \infty} \frac1{n} \sum_{(i, j) = (1, 1)}^{(n, n)} \frac1{i + j}$$

That one

Chris's sis

@BalarkaSen well, this one can be proved in many ways. Sure, I wanna see.

Balarka Sen

@Chris'ssis here

oops wrong link

corrected it

Chris's sis

@BalarkaSen there are 2 solutions

Balarka Sen

7:17 PM

Look at Euge's solution.

Huy

@BalarkaSen: Isn't that the standard way?

Balarka Sen

@Huy Which one are you referring to?

Huy

The one with Riemann sums.

Euge's

Balarka Sen

I dunno.

@Chris'ssis did it by Stolz-Cesaro

@rehband did it by some interesting integral trick

Huy

I see. I think we had it as an exercise in calculus 2 and did it with Riemann sums.

Balarka Sen

7:19 PM

It never crossed my mind to do it that way

topper

Looking to integrate $$\int_ \! 2\sqrt{x+1}+2\sqrt{x}\, \mathrm{d}x.$$ This is after multiplying by the conjugate to remove the denominator. I'm planning to split it due to the addition, and integrate both terms with substitution. Is there a smarter / more logical way?

Balarka Sen

I did it numbert theoretically, @Huy

Look at my solution.

Chris's sis

@BalarkaSen But I also showed you that solution, isn't it? Riemann sums ...

Nick

@BalarkaSen: I'm bad with many sums. Can you try $$ 1 + (1+2) + (1+2+3) + \dots + (1+2+3+\dots +n)$$

Balarka Sen

@Nick $$\sum_{n = 1}^\infty \frac{n(n+1)}{2}$$

Use Faulhaber's formulas.

Huy

7:20 PM

@topper: Well, substitution is a bit of overkill. You can just split due to addition and then integrate directly.

Balarka Sen

@Chris'ssis I thought you used Stolz-Cesaro?

Huy

@topper: Recall $\frac{\mathrm d}{\mathrm dx} x^n = n x^{n-1}$.

Chris's sis

@BalarkaSen I did it both by Cesaro and Riemann sums.

Balarka Sen

@Chris'ssis Ah, you never showed them to me.

Nick

@BalarkaSen: Omg, that's cool.

topper

7:21 PM

@Huy Of course. At least I knew it smelt bad enough to raise it. Of course the 2s are just products, then I use the product rule

Chris's sis

@BalarkaSen Anyway, thank you for telling me that! :-)

topper

I'm opening myself up here, but have people come in here asking thicker questions than me? Because everyone else here seems to live and breathe this stuff!

Huy

@topper: Not products. The factor rule. $\frac{\mathrm d}{\mathrm dx} a f(x) = a \frac{\mathrm d}{\mathrm dx} f(x)$.

Balarka Sen

@Chris'ssis And you don't mind since I posed it as a challenge problem did you?

topper

@Huy That's what I meant, power rule, call it what you will. That was a verbal typo, or... vypo

Huy

7:23 PM

@topper: Okay.

rehband

@Hippalectryon Hahahha

Chris's sis

@BalarkaSen No, not at all. ;)

Balarka Sen

I gave credits to you as an anonymous MSE user, btw. Is that fine?

Huy

@topper: The power rule is the first one I mentioned, the last one is called factor rule, methinks.

Hippalectryon

@rehband for what ?

rehband

7:23 PM

@Hippalectryon That pic you posted earlier

Hippalectryon

Ah yeah xD

topper

@Huy Yes, we're talking at cross-purposes. Basically I get it :)

Chris's sis

@Hippalectryon hahahahaha, that's crazy! :-)))))

Hippalectryon

I should try that with the TOS

Nick

@topper: Dude, the dx and the integral distributes to the terms, just integrate. Hint:$$ \int{\sqrt x} dx= \frac{2{x}^{3/2}}{3} +C$$

Balarka Sen

7:25 PM

What's so funny about that pic, @Hippa?

@Nick better.

I don't get it.

Yeah, that's not right @Nick. Also, a constant is missing. :P

Hippalectryon

@BalarkaSen It just feels akward

Balarka Sen

7:26 PM

Oh wait I just zoomed it

HAHAHAHAHHA

Hippalectryon

:D

rehband

Hahaha

How do you make those @ Hippalectryon?

Balarka Sen

HAHAHAHAH

This is awesome

Nick

@robjohn, @huy: Brain not working. Latex so hard. Phone so cwappy. End of excuses.

Hippalectryon

@rehband It's maaaaagic

rehband

7:29 PM

@Hippalectryon :O :O :O

topper

@Nick Does that work for anything in the root, even if let's say there was a quadratic in there, that it just moves to where the x is on the left? I never have to use substitution even if there's a more complicated expression in the root?

Huy

@topper: Think of integration as "anti-differentiation". Just find the derivative of $\sqrt{x+1}$ or even $\sqrt{x+c}$ for any $c \in \mathbb{R}$ and you'll see the constant doesn't really matter.

Nick

@topper: I need you think about anti-differentiating. Stop trying to integrate!

robjohn

@Nick $\frac{n(n+1)(n+2)}{6}$

Mike Miller

@Nick Never in my life have I seen someone denote a matrix with \mathbb, or really anything other than just italics...

Balarka Sen

7:31 PM

@robjohn Wasn't my question.

topper

:-O

Nick

@MikeMiller: I have. My life is weird.

Huy

@Nick: Where?

Nick

@robjohn: After factorizing, yeah, that's what I got! :D

@Huy: Here. Once upon a time.

robjohn

@Nick $$\sum_{n=k}^m\binom{n}{k}=\binom{m+1}{k+1}$$

2

Nick

7:38 PM

@robjohn: That's nicer :D Thank you.

topper

RIGHT. If I have time at the end of this exam tomorrow I am going to check every single plus and minus

hmph

Nick

@topper: You've gone from limits to integration in such a short time. How much time were you given too prepare, actually?

topper

@Nick I've had a couple of months but it's been a rough period, partly my fault, and partly things outside my control. And this is my first real studying apart from GMAT in 15 years, I think I've had to relearn how to learn

Basically I've learned a semester of material with no support, in a couple of months, with online videos. But I waited too long before I started doing questions. Too passive

And I play guitar well, I know how to learn, I just didn't apply the same principles here

Nick

@topper: Glad your making progress though. That's what matters right now.

topper

:) Thanks, and yes indeed I am. You lot don't hear about the questions I get right ;)

Nick

7:46 PM

@topper: :D Keep practicing, bud. Speed is crucial. Thinking is elemental.

Khallil

Let me know when you're ready, @Huy. ^_^

topper

I think I know what's confusing me. Take an expression like $(3-2x)^2$. To differentiate that, I know I have to use the chain rule. So when I see it in an integral I automatically think I have to substitute

Hippalectryon

is mathworld.wolfram.com down ?

Nick

@Hippalectryon: there are websites with robots that answer such questions, we are only mortals in this room.

rehband

@Hippalectryon Not working for me either

Nick

7:53 PM

@topper: To solve $\int (3-2x)^2 \cdot \text{d}x$, just use the binomial theorem. The question turns to integrable form. In most of the initial exercises of indefinite integration, expanding functions into workable components is all you have to do.

Khallil

You can make a substitution, @topper. $u=3-2x \implies \text{d}x = \frac{-1}{2} \text{ d}u$.

topper

@Hippalectryon downforeveryoneorjustme.com/mathworld.wolfram.com

Hippalectryon

@BalarkaSen In your limit of the sum of 1/nCr(2014,2014+k), how id you compute the Beta term ?

@topper @rehband thanks

topper

Sorry, I meant $(3-2x)^{-2}$ (power of minus two)

Hippalectryon

@BalarkaSen Oh wait nvm, it converges -> 0

topper

7:55 PM

@Hippalectryon I may not be able to help with maths, but I can do other geeky stuff

Khallil

The most important thing is to choose methods that you feel comfortable with during exams. Outside of exams, try to expand your set of tools.

Nick

@topper: Then do what @Khallil said or use partial fractions.

@Khallil: As I said, autocorrect.

Khallil

48 mins ago, by Daniel Fischer

@Nick Isn't there a way to turn off autocorrection?

topper

Ah, permalinking. Nice. So how come in chat.stackexchange.com/transcript/message/17509474#17509474 everything in the root (or the power to a half if you want to see it that way), there is no need to use substiution, , but with $(3-2x)^{-2}$ there is a need to do so?

Hmm. I thought that would paste the linked message in mine like @Khallil did ^^. How did you do that?

Khallil

Nick

8:01 PM

@Khallil: Yes.

@topper: notice that x and 1/x have different natures.

topper

G-d this is fascinating, the more I get into it

Khallil

Could you make your question a bit clearer, @topper?

What's G-d, as well?

Mike Miller

If anyone here likes groups, I have a group theory question for them.

topper

@Khallil I am not religious but judaism.about.com/od/judaismbasics/a/… is an old habit

Nick

@topper: Usually, if $\frac{d}{dx} \phi (x) = f(x)$, then $$\int f(ax + b) \cdot \text{d}x = \frac{1}{a} \phi(ax+b) + C$$ Hopefully, this obvious fact is helpful.

Khallil

8:05 PM

Oh, I see. Yea, we've all developed habits like that along the line. I developed a few of my weird ones when I was young.

topper

@Nick Helpful enough that it's pasted in my notes

Nick

@topper: Yeah, it's sort of obvious when you think about it interms of reversing differentiation. Thinking like this for basic problems helps relieve the stress of unnecessary substitution.

topper

(Which is easier to do when you've got a really good handle on differentiation.)

But I do enjoy how eventually the building blocks all relate to each other and come together

Khallil

I remember reading somewhere, that integration isn't the reverse process of differentiation.

Is that right?

Nick

@Khallil: Math is so twisted. lol

Khallil

8:13 PM

I have a limit question that you might be able to help me with, @topper and @Nick.

Nick

@Khallil: fire! :D

Khallil

I can't use L'Hôp, since $\left|\sin \frac{n\pi}{7}\right| \leqslant 1$.

I don't really want to use a series expansion for the sine function either.

Nick

@Khallil: It's probably $ \pi / 7$ if my math is right.

I could be wrong.

Khallil

Let me try it with the series expansion to check.

rehband

@Khallil The limit is simply $0$. The numerator $\sin(...) \leq 1$ and the denominator blows up :)

Khallil

8:18 PM

...

It's late and I'm FIFA deprived.

Nick

@rehband: Oh, the pi was in the sin() .. Yeah, you're right.

Khallil

It wouldn't matter if it wasn't, @Nick.

rehband

All good :)

Nick

@Khallil: Right. I get it. sin is only in [0,1], the denominator is $\infty$

Khallil

Exactly.

Pedro Tamaroff

8:20 PM

Can someone complement my answer here?

Khallil

Well, the denominator isn't infinity, @Nick. It tends to infinity.

Hippalectryon

@PedroTamaroff What is the HG function ?

Nick

@Khallil: I define the lemniscate as being a variable which denotes something tending to infinity. Got a problem with that?

Khallil

Yes. What's a lemniscate, @Nick?

Nick

@Khallil: The $\infty$ symbol

Pedro Tamaroff

8:22 PM

@Hippalectryon Hypergeometric.

Hippalectryon

Oh I see

Khallil

Oh. Yea, I've got a problem with it, @Nick.

Pedro Tamaroff

@Khallil ORLY

Khallil

You what m($\Gamma(2) + \Gamma(3)$)?

^_^

Nick

@Khallil: Then you're just going to have to make this poor kid type $\lim_{x \to \infty} x$

Khallil

8:27 PM

What about $\displaystyle \lim_{x \to 0^{+}} \dfrac{1}{x}$?

Nick

@Khallil: What about it.

Khallil

Or even, $\displaystyle \lim_{\theta \to n\pi/2^{-}} \tan \theta$?

It's a (poor) joke. They're all equivalent, @Nick!

Nick

It's all what I said. $\equiv\lim_{x \to \infty} x$

@Khallil: I dislike oscillatory limits. They're annoying.

Khallil

Changing the topic so as to move past that terrible joke ... clementines.

I don't mind anything.

I just particularly dislike things that are boring.

Nick

$$y = \lim_{x \to \infty} \sin x$$

Huy

8:30 PM

@Khallil: Friend was over to play two more games of FIFA, and now I should go to bed.

Nick

How do you denote it's solution

Khallil

Yep, I know what you mean. As far as I know, they aren't defined.

You can only say that it lies in $[-1,1]$.

No problem, @Huy!

Nick

$$y \in [-1. 1]$$

Khallil

Maybe another time. ^_^

That's how I'd do it, @Nick.

Nick

@Khallil: mhh, I found a writing style today.

Khallil

8:32 PM

Actually, it doesn't even matter. It's not defined.

You did, @Nick?

How'd you do that? I haven't changed my style since I was about 14.

Nick

@Khallil: Well, I'm a beginner. I'm learning.

Khallil

What do you mean by writing style?

Nick

@Khallil: ... leave it. Do you know how to plot the graphs of inverse trig. functions?

Khallil

Reflect in the line $y=x$ so long as the original function's domain is restricted so that it is one-to-one, @Nick.

Nick

I got that part but how do I draw $1/ \arcsin x$

Khallil

8:37 PM

Use your brain.

When $\arcsin x$ gets very small whilst staying positive, $\dfrac{1}{\arcsin x}$ gets very ___ ?

Nick

I got the hyperbola! :D

Khallil

You did?

I don't think I've graphed it before.

You shouldn't have gotten a hyperbola.

Well, not strictly speaking anyway.

It's domain should only be $x \in \left[ -1, 1 \right]$ so it's restricted.

Hyperbolae are defined by the following equation. $$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$$

Balarka Sen

$\lim_{n \to \infty} \beta(\text{blah}, n) = 0$

@robjohn That's a consequence of telescoping, no?

$$\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$$

Summing through and telescoping, one gets the desired result.

@Khallil Hmm, like?

Hippalectryon

I luv telescopes :c

Anyone has a super-awesome-tip-that-solves-some-hard-exercises-in-just-a-few-lines tip to share ?

Balarka Sen

@Hippalectryon Too general. What's the exercise you have in mind?

Hippalectryon

8:51 PM

Exercises on limits, sequences, integrals, ....

A bit like @Chris'ssis 's tip with Bell numbers

Balarka Sen

Which one?

Hippalectryon

To simplify questions with $\sum_{k=0}^\infty\frac{k^n}{k!}$ for low $n$

By replacing them with bell numbers

Balarka Sen

Ah, I see.

I don't have any on mind for limits sequence or integrals, as I haven't done too much of them

Hippalectryon

Even if it's for other kind of exercises, i'll take it :)

Balarka Sen

Maybe : whenever you see a question asking something like "is (a very complicated expression with factorials)/(another complicated expression with factorials) an integer?" try thinking about a combinatorial interpretation?

Hippalectryon

8:58 PM

Could you give an example ?

Balarka Sen

well, here's one i did recently : prove that $(n^2)!/(n!)^2$ is always an integer.

it's not too hard though

Hippalectryon

$=\binom{2n}{n}\frac{(n^2)!}{(2n)!}$

Hence integer

:D

Balarka Sen

Yes, that's my proof.

There is another cool one

mathhelpboards.com/challenge-questions-puzzles-28/…

Hippalectryon

Bacterius -____-

Balarka Sen

heh

interesting username, isn't it?

Hippalectryon

9:04 PM

Next time i'll make an account called Phagocytus -___-

@BalarkaSen didn't you forget the case n=1 ?

for $n=1$, $k=\frac{(n^2)!}{(2n)!}=1/2$

Balarka Sen

I don't care about trivialities.

=P

Hippalectryon

-___________________-

Balarka Sen

.

robjohn

@BalarkaSen yep

@Hippalectryon That diverges if $n\ge0$.

Hippalectryon

@robjohn What about it ?

robjohn

9:12 PM

@Hippalectryon Did you mean to have $k!$ in the denominator or $n!$? Or did you mean to sum in $k$?

Hippalectryon

@robjohn It does diverge with $n$, it's normal. But for $n$ fixed, it converges

Oh wait

It's sum of $k$ of course

robjohn

@Hippalectryon Is that better?

Hippalectryon

$eB_n=\sum_{k=0}^\infty\frac{k^n}{k!}$

robjohn

@Hippalectryon so my correction was correct :-)

Hippalectryon

Oh yeah

robjohn

9:17 PM

@Hippalectryon No, I just changed the variable of summation to $k$.

topper

I'm still here people, just quietly getting some questions right. Apologies for being smug, I'm just excited

robjohn

@topper we'll alert the media...

topper

I thought this was the media

robjohn

@topper perhaps the ~~mediocre~~ exceptional.

Hippalectryon

@BalarkaSen $\sum_{k=0}^\infty\frac{k^n}{k!}=\frac{2n!}{\pi}\int_0^\pi e^{e^{\cos\theta}\cos(\sin t)}\sin\left[e^{\cos\theta}\sin(\sin\theta)\right]\sin(n\theta)\mathrm{d}\theta$ :D

Balarka Sen

9:40 PM

Looks disgusting

topper

I want to find the points where $y^2=2x$ and $x^2=2y$ meet. Algebraically, I got to $x=2$, but $x$ is also $0$. I squared the second equation to turn it to $y^2=\frac{x^4}{4}$ then I had $2x = \frac{x^4}{4}$. What did I miss?

As in, how did I miss $x=0$ which I see intuitively, but I don't want to rely on intuition under pressure until I develop it better...

Khallil

Can you develop intuition?

Anyhow, algebraically, you can substitute the second equality into the first.

Hippalectryon

@topper If $y>0$ : $\begin{cases}y=\sqrt{2x}\\y=\frac{x^2}2\end{cases}$ hence $\sqrt{2x}=x^2/2$ thus $2x=x^4/2$ hence $x=0$ or $x=\sqrt[3]{4}$

topper

I guess I went from $x^4-8x=0$ to $x^3-8=0$, but I should have gone to $x(x^3-8)=0$. Just don't see what happened to that extra $x$

Khallil

Yea. You can't ensure that $x \neq 0$, so dividing through by $x$ could be fatal!

topper

9:50 PM

@Khallil Perhaps I'm not talking about intuition in its truest sense

@Khallil Great insight, thank you!

Khallil

Just like you wouldn't divide $\sin (\theta) f(\theta) = 0$ by $\sin \theta$ when solving some trigonometric equality, unless you've been given the condition that $\sin \theta \neq 0$ (or equally, $x\neq n\pi$ where $n \in \mathbb{N}$).

Hippalectryon

Above i meant at the end $x=\sqrt[3]{8}=2$, because I forgot to square the $2$ at the denominator at the beginning

Khallil

Thanks for the compliment, @topper!

What would you do if you didn't have math, @Chris'ssis?

Hippalectryon

Create them immediately ?

Chris's sis

@Khallil I'd try to invent / create it.

Khallil

9:57 PM

Isn't math discovered and not created?